Binary mode or Multi-label mode is correct when using binary crossentropy and sigmoid output function on multi-label classification
Since you have a multiclass classification problem rather than a binary classification problem (i.e. a two-class problem), I recommend to adjust your architecture and use softmax instead of sigmoid as final activation function and categorcal_crossentropy instead of binary_crossentropy.
Softmax will ensure all your outputs are valid probabilities. This is crucial in calculating the loss of your model. Using sigmoid will only ensure this if you have binary classes. Further categorical crossentropy is used for multiclass problems, again this is to make sure you calculate the correct loss. This is just a general model characteristic:
binary -&gt; sigmoid &amp; binary_crossentropy
multi-class -&gt; softmax &amp; (sparse_)categorical_crossentropy
To answer you original question, I think you should use average = micro, macro or weighted when calculating the F1-score. This depends on what you like your model to put more emphasis on and how severe your class imbalances are.